Systems and methods for OFDM transmission and reception

ABSTRACT

A discrete cosine transform (DCT)-based orthogonal frequency-division multiplexing (OFDM) system is provided with a zero-padding guard interval and MMSE reception. The performance of the DCT-OFDM system with the zero-padding guard interval scheme and the minimum mean-square error (MMSE) receiver over time-varying multipath Rayleigh fading channels is investigated. The results show that employing the proposed DCT-OFDM system rather than the conventional DFT-OFDM system can provide better bit error rate performance in practical systems, by as much as 6 dB in signal-to-noise ratio.

This application is the National Phase of International Application No.PCT/CA2005/001560 filed on Oct. 13, 2005, which claims priority fromU.S. Provisional application No. 60/617,637, which documents are bothincorporated herein by reference in their entirety.

FIELD OF THE INVENTION

The invention relates to systems and methods for OFDM transmission andreception.

BACKGROUND OF THE INVENTION

Orthogonal frequency-division multiplexing (OFDM) is being widely usedin the physical layer specifications, such as IEEE 802.11a, IEEE802.16a, and HIPERLAN/2, of many broadband wireless access systems.Particularly, IEEE 802.11a, providing data rates ranging from 6 Mbits/sto 54 Mbits/s by a link adaptation scheme with different modulationformats from binary phase shift keying (BPSK) to 64-ary quadratureamplitude modulation (64-QAM), has become a popular computer industrystandard for wireless network cards. It is also being used in digitalaudio broadcasting (DAB) and digital video broadcasting (DVB) systems inEurope. Moreover, AT&T Labs recently demonstrated theirfourth-generation (4G) system with OFDM technology for the downlink.OFDM technology can find its way to these applications because of itswideband nature and the ability to effectively convert a frequencyselective fading channel into several nearly flat fading channels.

In discrete Fourier transform (DFT) OFDM, intersymbol interference (ISI)can be completely canceled by using the cyclic prefix (CP) schemeprovided the CP is longer than the channel impulse response (CIR). SeeA. Peled and A. Ruiz, “Frequency domain data transmission using reducedcomputational complexity algorithms,” in Proc. IEEE Int. Conf. Acoust.,Speech, Signal Processing, 1980, pp. 964-967. In addition, if thecarrier frequency is synchronized perfectly, and the CIR does not varyin one OFDM frame, there will be no intercarrier interference (ICI)components in the received signals. Therefore, one can use a one-tapequalizer in the receiver to compensate the channel distortion.Actually, the CP scheme works because of the fact that the DFT of thecircular convolution of two sequences is equal to the multiplication ofthe DFTs of these two sequences. See A. V. Oppenheim, R. W. Schafer,with J. R. Buck, Discrete-Time Signal Processing, 2nd Ed., PrenticeHall, 1998. DFT-OFDM can not guarantee symbol recovery if the channeltransfer function has zero(s) on the FFT grid.

As an alternative to use of the CP, a zero padding (ZP) scheme can beused in DFT-OFDM. A ZP scheme can provide ISI-free transmission if thelength of channel impulse response is less than the length of the ZP.This scheme ensures symbol recovery regardless of the channel zerolocations.

The above mentioned systems are discrete Fourier transform (DFT)-basedmulticarrier modulations (MCMs) where the complex exponential functionsset is employed as an orthogonal basis.

An alternative to a DFT-based OFDM system is a system employing anotherorthogonal basis, namely a single set of cosinusoidal functionscos(2πnF_(Δ)t) where n=0, 1, . . . , N−1 and 0≦t<T, to implement themulticarrier modulation scheme. The minimum F_(Δ) required to satisfythe orthogonality condition

$\begin{matrix}{{\int_{0}^{T}{\sqrt{\frac{2}{T}}{\cos\left( {2\pi\; k\; F_{\Delta}t} \right)}\sqrt{\frac{2}{T}}{\cos\left( {2\pi\; m\; F_{\Delta}t} \right)}{\mathbb{d}t}}} = \left\{ \begin{matrix}{1,} & {k = m} \\{0,} & {k \neq m}\end{matrix} \right.} & (1)\end{matrix}$is ½ T Hz. This scheme can be synthesized using a discrete cosinetransform (DCT). This scheme will be denoted as DCT-OFDM, and theconventional OFDM system as DFT-OFDM. As far as fast implementationalgorithms are concerned, the fast DCT algorithms proposed in W. H.Chen, C. H. Smith, and S. C. Fralick; “A fast computational algorithmfor the discrete cosine transform,” IEEE Trans. Commun., vol. 25, pp.1004-1009, September 1977 and Zhongde Wang, “Fast algorithms for thediscrete w transform and for the discrete Fourier transform,” IEEETrans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 803-816,August 1984 could provide fewer computational steps than fast Fouriertransform (FFT) algorithms.

Unfortunately, in general, the DCT does not have the circularconvolution multiplication property. Inserting the CP scheme directlyinto the proposed DCT-OFDM system does eliminate ISI. However, as a sideeffect, ICI is introduced even when the CIR does not change in oneDCT-OFDM frame.

In K. R. Rao, P. Yip, Discrete Cosine Transform. Academic Press, 1990and S. A. Martucci, “Symmetric convolution and the discrete sine andcosine transform,” IEEE Trans. Signal Processing, vol. 42, pp.1038-1051, May 1994 it was suggested that when the given sequences areevenly symmetrically extended, a circular convolution property similarto the DFT can be found. By employing this property, a DCT-based OFDMsystem is proposed in G. D. Mandyam, “On the discrete cosine transformand OFDM systems,” in Proc. ICASSP 2003, pp. 544-547. However, thesymmetric extension of the original data sequence reduces the datatransmission efficiency by at least one-half.

SUMMARY OF THE INVENTION

According to one broad aspect, the invention provides an OFDMtransmitter adapted to, during each OFDM symbol interval, map datasymbols to orthogonal sub-carriers of the form cos(2πnF_(Δ)t) where n=0,1, . . . , N−1 and 0≦t<T, and to insert zeros after each OFDM symbolinterval.

In some embodiments, a system comprises the OFDM transmitter and areceiver adapted to, during each OFDM symbol interval, recover estimatesof the data symbols transmitted by the OFDM transmitter.

In some embodiments, the receiver is adapted to perform MMSE estimationto recover the estimates.

In some embodiments, the OFDM transmitter comprises: an inverse discretecosine transform that produces a time domain output for each OFDM symbolinterval in which data symbols are mapped to the sub-carriers; and azero padding function adapted to insert zeros after each OFDM symbolinterval.

In some embodiments, the inverse discrete cosine transform is an N-pointIDCT that produces N time domain samples; the zero padding functioninserts G zeros after each set of N time domain samples, where G isselected to substantially eliminate ISI between consecutive OFDM symbolsafter transmission over a channel.

In some embodiments, G is selected to be greater than or equal to alength of a channel impulse response for the channel.

In some embodiments, the receiver performs MMSE estimation in accordancewith:

$\begin{matrix}{{\hat{X}(i)}\begin{matrix}{= {D\;{H^{H}\left( {{H\; H^{H}} + {\sigma^{2}I_{M}}} \right)}^{- 1}{\overset{\rightharpoonup}{R}(i)}}} \\{= {{D\left( {{H^{H}H} + {\sigma^{2}I_{N}}} \right)}^{- 1}H^{H}{\overset{\rightharpoonup}{R}(i)}}}\end{matrix}} & (13)\end{matrix}$where I_(M) is an M×M identity matrix, {circumflex over (X)}(i) is theMMSE estimate, H=H₀C_(zp), where H₀ is a channel matrix, and C_(zp) is azero padding matrix, D is a matrix representation of the discrete cosinetransform, M is a sequence length including zero padding, and N is asequence length not including zero padding, (*)^(H) is a conjugatetranspose operation, and σ² is an additive white noise Gaussian noisevariance.

In some embodiments, the system comprises a one dimensional modulationand demodulation scheme.

In some embodiments, the system comprises a two dimensional modulationand demodulation scheme.

According to another broad aspect, the invention provides a methodcomprising: during each OFDM symbol interval, mapping data symbols toorthogonal sub-carriers of the form cos(2πnF_(Δ)t) where n=0, 1, . . . ,N−1 and 0≦t<T; and inserting zeros between consecutive OFDM symbolintervals and transmitting a resulting signal.

In some embodiments, the method further comprises: at a receiver,receiving the resulting signal over a channel, and during each OFDMsymbol interval processing the resulting signal received over thechannel to recover estimates of the data symbols.

In some embodiments, processing comprises performing MMSE estimation.

In some embodiments, mapping comprises performing an inverse discretecosine transform to produce a time domain output for each OFDM symbolinterval in which data symbols are mapped to the sub-carriers.

In some embodiments, the inverse discrete cosine transform is an N-pointIDCT that produces N time domain samples; inserting zeros comprisesinserting G zeros after each set of N time domain samples, where G isselected to substantially eliminate ISI between adjacent OFDM symbolsafter transmission over the channel.

In some embodiments, the MMSE estimation is performed in accordancewith:

$\begin{matrix}{{\hat{X}(i)}\begin{matrix}{= {D\;{H^{H}\left( {{H\; H^{H}} + {\sigma^{2}I_{M}}} \right)}^{- 1}{\overset{\rightharpoonup}{R}(i)}}} \\{= {{D\left( {{H^{H}H} + {\sigma^{2}I_{N}}} \right)}^{- 1}H^{H}{\overset{\rightharpoonup}{R}(i)}}}\end{matrix}} & (13)\end{matrix}$where I_(M) is an M×M identity matrix, {circumflex over (X)}(i) is theMMSE estimate, H=H₀C_(zp), where H₀ is a channel matrix, and C_(zp) is azero padding matrix, D is a matrix representation of the discrete cosinetransform, M is a sequence length including zero padding, and N is asequence length not including zero padding, (*)^(H) is a conjugatetranspose operation, and σ² is an additive white noise Gaussian noisevariance.

In some embodiments, the method further comprises: performing onedimensional modulation and demodulation.

In some embodiments, the system comprises: performing two dimensionalmodulation and demodulation.

According to another broad aspect, the invention provides a receiveradapted to, during each OFDM symbol interval: receive a signalcomprising data symbols mapped to orthogonal sub-carriers of the formcos(2πnF_(Δ)t) where n=0, 1, . . . , N−1 and 0≦t<T, and having a zeroguard band between consecutive symbol intervals, the guard band having aduration selected to substantially eliminate ISI between consecutivesymbols after transmission over a channel; and during each OFDM symbolinterval, perform MMSE estimation to recover estimates of the datasymbols.

In some embodiments, the receiver is adapted to perform demodulation ofa one dimensional modulation scheme.

In some embodiments, the receiver is adapted to perform demodulation ofa two dimensional modulation scheme.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a DCT-based orthogonal frequency-divisionmultiplexing system;

FIG. 2 is a block diagram of a discrete-time baseband model of the MMSEreceiver for a zero-padded DCT-OFDM system provided by an embodiment ofthe invention;

FIG. 3 is a graph of BER performances of the MMSE receiver with perfectchannel information for 64-subcarrier DCT-OFDM and 64-subcarrierDFT-OFDM, both with BPSK modulation, over a time-varying multipathRayleigh fading channel;

FIG. 4 is a graph of BER performances of the MMSE receiver with perfectchannel information for a 64-subcarrier DCT-OFDM and a 64-subcarrierDFT-OFDM, both with BPSK modulation, over a time-varying multipathRayleigh fading channel;

FIG. 5 is a graph of BER performances of the MMSE receiver with channelestimation for a 64-subcarrier DCT-OFDM and a 64-subcarrier DFT-OFDM,both with QPSK, over a time-varying multipath Rayleigh fading channel;

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The intercarrier interference (ICI) analysis and the exact bit errorrate performance comparisons in the presence of frequency offset betweenthe DCT-OFDM system and the DFT-OFDM system in applicants paper P. Tanand N. C. Beaulieu, “Precise bit error probability analysis of DCT OFDMin the presence of carrier frequency offset on AWGN channels,” to appearin GLOBECOM05, Nov. 28-Dec. 2, 2005 hereby incorporated by reference inits entirety, have shown that the DCT-OFDM scheme can be more robust tofrequency offset than the conventional DFT-OFDM scheme due to the energycompaction property of the DCT described in A. V. Oppenheim, R. W.Schafer, with J. R. Buck, Discrete-Time Signal Processing, 2nd Ed.,Prentice Hall, 1998; that is, the signal energy is concentrated in a fewlow-index DCT coefficients while the remaining coefficients are zero orare negligibly small. In this regard, it has been shown that the DCT isclose to optimal in terms of energy compaction capabilities. See K. R.Rao, P. Yip, Discrete Cosine Transform. Academic Press, 1990. In otherstudies about DCT-OFDM, it is proposed that by feeding a symmetricallyextended sequence into a DCT-OFDM system, in the case of static andexponentially decaying channel profiles with intersymbol interference(ISI), the throughput lower bound of the DCT-based OFDM system is higherthan that of the DFT-based OFDM system. See G. D. Mandyam, “On thediscrete cosine transform and OFDM systems,” in Proc. ICASSP 2003, 2003,pp. 544-547. As a side effect, the data transmission efficiency of thatscheme will be decreased by at least one-half. In addition, the staticchannel assumption usually does not hold in real wireless transmissionenvironments. A more meaningful performance measure in practical digitalwireless communications systems, bit error rate (BER), is not examinedin that work. Considering there is a bandwidth advantage for a DCT-basedsystem, in J. Tan and G. L. Stüber, “Constant envelope multi-carriermodulation,” in Proc. MILCOM 2002, vol. 1, pp. 607-611, 2002 a DCT,rather than a DFT, was used to implement multicarrier modulation intheir constant envelope MCM system. Based on the same idea, F. Xiong,“M-ary amplitude shift keying OFDM system,” IEEE Trans. Commun., vol.51, pp. 1638-1642, October 2003, it was proposed to use a coherent√{square root over (M)}-ary amplitude shift keying OFDM system insteadof the M-ary QAM OFDM system.

An embodiment of the invention provides a DCT-OFDM system that featuresthe use of a zero-padding guard interval scheme employed in combinationwith a minimum mean-square error (MMSE) receiver. In someimplementations, the new scheme can not only eliminate the ISI but alsoavoid decreasing the transmission efficiency.

System Model

In a DCT-OFDM system, input bits are first mapped into symbols based ona specific signaling format in a data encoder. Then N symbols areserial-to-parallel (S/P) converted into N low-rate data streams. The ithparallel symbol block is denoted as a vector{right arrow over (X)}(i)=[d ₀ ^(i) d ₁ ^(i) . . . d _(N-1)^(i)]^(T)  (2)where [x]^(T) denotes transposition. With the introduction of azero-padding interval longer than the channel impulse response, theintersymbol interference (ISI) or inter-block interference (IBI) in OFDMsystem can usually be ignored. For the sake of notational convenience,the superscript i is temporarily omitted since only the ith data blockis considered currently. The N symbols {d_(k)}_(k=0) ^(N-1), each withtime duration T, are then multiplexed by modulating correspondingsubcarriers. Unlike conventional OFDM, here the set cos(2πnF_(Δ)t), n=0,. . . , N−1 and 0<=t<T is used as subcarriers to implement themulticarrier modulation. To maintain the orthogonality, the subcarriersneed a minimum frequency spacing F_(Δ) of ½ T Hz. The multiplexed signalx(t) can be written as

$\begin{matrix}{{{x(t)} = {\sum\limits_{n = 0}^{N - 1}{d_{n}g_{n}{\cos\left( {n\;\pi\;{t/T}} \right)}}}}{where}} & \left( {3a} \right) \\{g_{n} = \left\{ \begin{matrix}{\sqrt{1/N},} & {n = 0} \\{\sqrt{2/N},} & {{n = 1},2,\ldots\mspace{11mu},{N - 1.}}\end{matrix} \right.} & \left( {3b} \right)\end{matrix}$

Sampling the continuous time signal x(t) at time instants t_(m) where

$\begin{matrix}{{t_{m} = \frac{T\left( {{2m} + 1} \right)}{2N}},{m = 0},1,\ldots\mspace{11mu},{N - 1}} & (4)\end{matrix}$gives a discrete sequence

$\begin{matrix}{x_{m} = {\sqrt{\frac{2}{N}}{\sum\limits_{n = 0}^{N - 1}{d_{n}\beta_{n}{\cos\left( \frac{\pi\;{n\left( {{2m} + 1} \right)}}{2N} \right)}}}}} & \left( {5a} \right)\end{matrix}$where

$\begin{matrix}{\beta_{n} = \left\{ \begin{matrix}{{1/\sqrt{2}},} & {n = 0} \\{1,} & {{n = 1},2,\ldots\mspace{11mu},{N - 1.}}\end{matrix} \right.} & \left( {5b} \right)\end{matrix}$Eq. (5a) is just the inverse discrete cosine transform (IDCT) which isreferred to as IDCT-2 A. V. Oppenheim, R. W. Schafer, with J. R. Buck,Discrete-Time Signal Processing, 2nd Ed., Prentice Hall, 1998. Thediscrete cosine transform (DCT) will restore the original signal d_(n)as

$\begin{matrix}{d_{n} = {\sqrt{\frac{2}{N}}\beta_{n}{\sum\limits_{m = 0}^{N - 1}{x_{m}{{\cos\left( \frac{\pi\;{n\left( {{2m} + 1} \right)}}{2N} \right)}.}}}}} & (6)\end{matrix}$

FIG. 1 is a detailed block diagram for a DCT-OFDM system includingin-phase and quadrature modulators in the presence of carrier frequencyoffset Δf and phase error φ. Because the DCT is a real transform, it isnot necessary to use the quadrature modulator for one-dimensionalsignaling formats, such as BPSK and M-ary pulse amplitude modulation(M-PAM).

Input data 100 is converted to parallel form with serial-to-parallelconverter 101. The output of this is processed by IDCT 102 and convertedback to serial form at 104. In-phase and quadrature components arefiltered in respective low pass filters 106, 108 and then modulatedusing in-phase and quadrature modulators 110, 112. The output of themodulators is summed at 114 and bandpass filters at 116 and thentransmitted over channel 118 where it is assumed that additive whiteGaussian noise is added at 120. At the receiver, a received signal isbandpass filtered at 122, demodulated with in-phase and quadraturedemodulators 124, 126, low pass filtered at 128, 130 and then sampled at132. The sampled output is then converted to parallel form at 134,processed by DCT function 136 and converted back to serial form at 138.Functional element 140 represents any decision process taken on theoutput of the DCT process, and the overall data output is indicated at142.

Performance over Time-Varying Multipath Rayleigh Fading Channels

Multipath channels will introduce inter-symbol interference (ISI) whichcan be mitigated by employing a proper ISI-cancellation scheme. Rapidchange of channel states, modeled by the Doppler frequency shift, willdestroy the orthogonality between subcarriers, and thus gives ICI.Therefore, an equalization scheme should be employed to suppress theICI.

According to a preferred embodiment of the invention, a zero-paddingscheme is employed in a DCT-OFDM system to mitigate ISI, and a MMSEequalizer is used in the receiver to reduce ICI.

Zero-Padding DCT-OFDM

A new zero-padding scheme is provided that can eliminate the ISI andavoid decreasing the transmission efficiency as well.

Let an N×1 vector {right arrow over (X)}(i) represent a length N datablock transmitted in the ith DCT-OFDM symbol. The N×1 vector {rightarrow over (Y)}(i) represents the DCT of {right arrow over (X)}(i). Onemay write{right arrow over (Y)}(i)=D ^(T) {right arrow over (X)}(i)  (7)where D^(T) is the IDCT matrix and D is the DCT matrix for a length-Nsequence. The rth (0≦r≦N−1) row and cth (0≦c≦N−1) column element D_(r,c)of the DCT matrix D is defined as

$\begin{matrix}{D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}.}}} & (8)\end{matrix}$According to an embodiment of the invention, a length G zero-paddingsequence is added after sequence {right arrow over (Y)}(i). The newsequence {right arrow over (Y)}′(i) will become a zero-padded sequenceof length M (M=N+G) given as{right arrow over (Y)}(i)=C _(zp) {right arrow over (Y)}(i)  (9)where C_(zp)=[I_(N) 0_(N×G)]^(T) is an M×N zero-padding matrix, andI_(N) is an N×N identity matrix.

Under a wide-sense stationary uncorrelated scattering (WSSUS)assumption, a frequency selective multipath fading channel can bemodeled as a tapped-delay-line (TDL) with L+1 time-varying coefficientsh₀(t), h₁(t), and h_(L)(t).

In preferred implementations, G is selected to be greater than or equalto L, such that there will be no ISI components in the received M×1fading signal plus noise vector

$\begin{matrix}\begin{matrix}{{\overset{\_}{R}(i)} = {{H_{0}C_{zp}{\overset{\_}{Y}(i)}} + {\overset{\rightharpoonup}{W}(i)}}} \\{= {{H{\overset{\rightharpoonup}{Y}(i)}} + {\overset{\rightharpoonup}{W}(i)}}}\end{matrix} & (10)\end{matrix}$where H₀ is the M×M channel convolutional matrix whose element in therth (0≦r≦M−1) row and cth (0≦c≦M−1) column is

$\begin{matrix}{{H_{0}\left( {r,c} \right)} = \left\{ \begin{matrix}{{h_{r - c}(t)},} & {0 \leq {r - c} \leq L} \\{0,} & {otherwise}\end{matrix} \right.} & (11)\end{matrix}$and {right arrow over (W)}(i) is the AWGN vector with mean zero andvariance σ², and where the matrix H is determined by H₀C_(zp). L canhave many values depending on the propagation environment and the datatransmission rate. Values from L=3 to L=8 are typical, for current datatransmission rates and channels, but L could be 0 and L could be largefor some environments and some transmission rates, say L=hundreds. Inthe HIPERLAN/2 channel model, one might for example choose G to begreater than L=8 for ZP DCT-OFDM.

In the absence of noise, the zero-forcing equalizer will recover thesignal completely according to{circumflex over (X)}(i)=DH ^(†) {right arrow over (R)}(i)  (12)where H^(†) is the pseudo-inverse of matrix H, H^(†)=(H^(H)H)⁻¹H^(H),and [x]^(H) denotes conjugate transposition. However, ignoring noise isnot practical.

In a preferred embodiment, a MMSE receiver is used to give optimaldecision variables, in the sense of minimizing the mean-square errorbetween {right arrow over (X)}(i) and {circumflex over (X)}(i), as

$\begin{matrix}\begin{matrix}{{\hat{X}({\mathbb{i}})} = {{{DH}^{H}\left( {{HH}^{H} + {\sigma^{2}I_{M}}} \right)}^{- 1}{\overset{\rightharpoonup}{R}({\mathbb{i}})}}} \\{= {{D\left( {{H^{H}H} + {\sigma^{2}I_{N}}} \right)}^{- 1}H^{H}{\overset{\rightharpoonup}{R}({\mathbb{i}})}}}\end{matrix} & (13)\end{matrix}$where I_(M) is an M×M identity matrix. For two-dimensional signalingformats, both the real and imaginary components of {circumflex over(X)}(i) will be used as the decision vector. For one-dimensionalsignaling formats, the real part of {circumflex over (X)}(i) will beused as the decision vector. A discrete-time baseband model of the MMSEreceiver for the zero-padded DCT-OFDM is shown in FIG. 2. Note that thediscrete cosine transform has been included in the minimum mean-squareerror estimation component at the receiver end, this being the operation“D” in equation 13.

An information source is indicated at 200. The output of this is subjectto coding at 202. The coded output is symbol mapped at 204 and thenprocessed by IDCT 206. The output of IDCT 206 is then zero padded at 208to produce the overall output that is then transmitted over afrequency/time selective fading channel with AWGN 210. At the receiver,channel estimation 212, MMSE 214, and AGC/coarse synchronization 216 areperformed. For a given implementation, one or more of channel estimationand AGC/coarse synchronization may be omitted. In the illustratedexample, the output of the MMSE 214 and channel estimation 212 is fed toa soft decision and de-mapping function 218 the output of which is fedto coding function 220 which produces a decoded output that is sent toan information sink 222. In an alternative, the output of the MMSE isused for hard decision decoding.

The functional elements of FIG. 2 can be implemented using anyappropriate technology. In one example implementation of a transmitterand receiver using zero padding, for symbols with M non-zero levels, anextra “zero level” is introduced. For example, in a system withnominally two levels {1A, −1A} the modified symbol set {+1A, −1A, 0} canbe used. Level 0 (the zero symbol) is transmitted when zeros are padded.Similarly, if the transmitter symbols are {−3A, −1A, +1A, +3A} on eachquadrature branch, the symbols {−3A, −1A, 0, +1A, +3A} can betransmitted including sending a symbol when padding a zero.

The following is a particular example of how the zero padding function208 might operate. The zero padding function 208 receives N paralleloutputs from the IDCT 206 over an OFDM symbol duration of T, andproduces a serial output of length N+G=M sequence over time T+delta,where delta=T*G/N. To achieve this, the zero padding function convertsthe N parallel outputs of the IDCT 206 into a serial output of length Nover time T, and then pads G zeros over time delta. Note that the IDCTinput symbol rate (the rate at which symbols are produced by the symbolmapping 204) also needs to be matched to the new symbol rate thataccounts for the zero padding. The system can accommodate an IDCT inputsymbol rate of N/(T+delta) symbols/s.

A very specific circuit for a DCT-OFDM system has been described. It isto be understood that many different implementations of such a systemare possible within the capabilities of one of ordinary skill of theart. More generally, any transmitter design that incorporates the zeropadding approach in conjunction with DCT-OFDM signalling iscontemplated. In the receiver, any receiver processing that takes intoaccount the zero padding of a transmitted signal, and performs MMSEestimation of DCT-OFDM signals is contemplated.

In another embodiment, a different receiver implementation may be used.Examples of implementations that might be appropriate that are providedin other embodiments include sphere decoding, V-BLAST and maximumlikelihood all of which are more complex and expensive than MMSE.

The data is recovered on the basis of the decision vector in systemswithout coding and in systems using hard decision decoding. In thelatter, the data are input to a decoding operation. In systems usingsoft decision decoding, the decision vector is input directly to adecoding operation.

Equation 13 provides a MMSE receiver for a very specific application. Itis to be understood that a receiver may implement this equation usingany number of available techniques. Examples include but are not limitedto hardware, software, ASICs, FPGAs, DSPs, or any combination of suchdevices. Also, depending upon assumptions made regarding the channel,and the nature of the signals transmitted, the MMSE receiver may takeother forms than that specifically disclosed in equation 13.

In equation 13, “D” represents the discrete cosine transform. Thisfunction can be implemented using any available technique. The matrix His determined by H₀C_(zp), C_(zp) being the zero padding matrix, and H₀being the channel convolution matrix. It is assumed that the channelimpulse response is known and that from this the channel convolutionmatrix can be generated. It is assumed that the channel estimate isobtained by other means. It is also noted that the form of the MMSEreceiver 13 implicitly builds in the stripping of the padded zeros.

Computer Simulations

The bit error rate performance of an uncoded DCT-OFDM system and anuncoded DFT-OFDM system over time-varying multipath Rayleigh fadingchannels is evaluated in this section. In the case of DFT-OFDM, the DCTmatrix D in Eq. (13) should be replaced with the unitary Fouriertransform matrix.

The HIPERLAN/2 channel model A defined in ETSI Normalization Committee,“Channel models for HIPERLAN/2 in different indoor scenarios,” EuropeanTelecommunications Standards Institute, Sophia-Antipolis, France,Document 3ERI085B, Mar. 30, 1998 is employed, which describes thetypical transmission environment for a large office withnon-line-of-sight propagation, in the simulations. The channel tapsα_(i)(t) are independent and identically distributed zero-mean complexGaussian random processes. They are generated independently withclassical 2-D isotropic scattering omnidirectional receiver antennaDoppler spectrum. The number of subcarriers in these simulations is 64,and the length of the zero-padding is G=16. Since L=8 in the T-spacedTDL model for HIPERLAN/2 channel model A, no ISI occurs in thesimulations. All of the simulations assume a fast fading channel, thatis, channel state information will change in one OFDM symbol. To ensurethe reliability of the computer simulations, 2¹⁷ OFDM symbols aregenerated to obtain each BER value in the simulations.

Under a perfect channel state information assumption, the BERperformances of the MMSE receiver for 64-subcarrier DCT-OFDM and64-subcarrier DFT-OFDM are compared, with BPSK modulation and QPSKmodulation, in FIG. 3 and FIG. 4, respectively. One can observe thatbetter BER performance can be achieved with the employment of theproposed DCT-OFDM scheme, rather than the DFT-OFDM scheme in FIG. 3.Particularly, in the case of f_(d)T=0.02, a 5.52 dB gain in SNR over theDFT-OFDM system at a bit error rate of 10⁻⁴ can be achieved by theDCT-OFDM system. In the case of QPSK modulation, in FIG. 4, the BERperformances for DCT-OFDM or DFT-OFDM are quite similar in the case off_(d)T=0.01 and 0.02. However, by using DCT-OFDM, rather than DFT-OFDMin both cases, one can achieve about 2.3 dB gain in SNR at a bit errorrate of 10⁻⁴.

Comparing FIG. 3 and FIG. 4, one can see that the BPSK DCT-OFDM achievesa significant performance improvement over the QPSK DCT-OFDM. The reasonfor this is that there exists interference between the in-phase signaland the quadrature signal in the case of QPSK DCT-OFDM but not in thecase of BPSK DCT-OFDM since the discrete cosine transform is a realtransform. However, for DFT-OFDM, cross-quadrature interference existsin both cases.

The fast fading channel estimation algorithm defined in Y. S. Choi, P.J. Voltz, and F. A. Cassara, “On channel estimation and detection formulticarrier signals in fast and selective Rayleigh fading channels,”IEEE Trans. Commun., vol. 49, pp. 1375-1387, August 2001, where thechannel state information of the middle K−1 symbols is obtained from Mpilot symbols on both sides by using a Wiener filter method, can be usedin the DCT-OFDM system. However, unlike in the above reference, here itis assumed that M pilot symbols are used for the estimation of thelatter K−1 symbols. This will reduce the delay introduced by the otheralgorithm, but greater estimation error may result in this case. FIG. 5shows the BER performance of the two systems when performing channelestimation with M=P K=1. An error rate floor can be observed in thisfigure. In particular, in the case of f_(d)T=0.01, DCT-OFDM outperformsDFT-OFDM by about 3.5 dB at a bit error rate of 2×10⁻³.

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

We claim:
 1. An OFDM transmitter comprising: an inverse discrete cosinetransform (IDCT) functional unit configured to produce a set of timedomain samples for each OFDM symbol interval in which data symbols aremapped to a set of N orthogonal sub-carriers of the form cos(2πnF_(Δ)t)where n=0, 1, . . . , N−1 and 0≦t<T, where N≧2, F_(Δ) is a frequencyspacing between subcarriers, T is a time duration of each OFDM symbolinterval, wherein the set of time domain symbols {right arrow over(Y)}(i) equal D^(T) X(i) where X(i) are the data symbols, and whereinD^(T) is the transpose of an N×N matrix with the element in the cth rowand the rth column defined by$D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}}$for 0≦r≦N−1, and 0≦c≦N−1, where $\beta_{r} = \left\{ {\begin{matrix}{{1/\sqrt{2}},} & \; & {r = 0} \\{1,} & {{r = 1},2,\ldots\mspace{14mu},} & {N - 1}\end{matrix};} \right.$ and a zero padding functional unit that for eachOFDM symbol interval is configured to insert a contiguous set of zerosin the time domain after the set of time domain samples; wherein theinverse discrete cosine transform functional unit is an N-point IDCTthat produces N time domain samples; the zero padding functional unit isconfigured to insert G zeros in the time domain after each set of N timedomain samples, where G is selected to substantially eliminateintersymbol interference (ISI) between consecutive OFDM symbols aftertransmission over a channel, to produce a zero-padded sequence equal to{right arrow over (Y)}′(i)=C_(zp){right arrow over (Y)}(i) whereC_(zp)=[I_(N) 0_(N×G)]^(T) is an M× N zero-padding matrix, and I_(N) isan N×N identity matrix, and O_(N×G) is an N×G matrix containing allzeros.
 2. A system comprising the OFDM transmitter of claim 1 and areceiver adapted to, during each OFDM symbol interval, recover estimatesof the data symbols transmitted by the OFDM transmitter.
 3. The systemof claim 2 wherein the receiver is adapted to perform MMSE estimation torecover the estimates.
 4. The system of claim 1 wherein G is selected tobe greater than or equal to a length of a channel impulse response forthe channel.
 5. The system of claim 3 wherein the receiver performs MMSEestimation in accordance with:{circumflex over (X)}(i)=DH ^(H)(HH ^(H)+σ² I _(M))⁻¹ {right arrow over(R)}(i)=D(H ^(H) H+σ ² I _(N))⁻¹ H ^(H) R(i)  (13) where I_(M) is an M×Midentity matrix, {circumflex over (X)}(i) is the MMSE estimate,H=H₀C_(zp), where H₀ is a channel matrix, and C_(zp) is a zero paddingmatrix, D is a matrix representation of the discrete cosine transform, Mis a sequence length including zero padding, and N is a sequence lengthnot including zero padding, (*)^(H) is a conjugate transpose operation,and σ² is an additive white noise Gaussian noise variance.
 6. The systemof claim 2 comprising a one dimensional modulation and demodulationscheme.
 7. The system of claim 2 comprising a two dimensional modulationand demodulation scheme.
 8. A method comprising: during each OFDM symbolinterval, performing an inverse discrete cosine transform (IDCT) toproduce a set of time domain samples to map data symbols to a set of Northogonal sub-carriers of the form cos(2πnF_(Δ)t) where n=0, 1, . . . ,N−1 and 0≦t<T, where N≧2, F_(Δ) is a frequency spacing betweensubcarriers, T is a time duration of each OFDM symbol interval, whereinthe set of time domain symbols {right arrow over (Y)}(i) equal D^(T)X(i) where X(i) are the data symbols, and wherein D^(T) is the transposeof an N×N matrix with the element in the cth row and the rth columndefined by$D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}}$or 0≦r≦N−1, and 0≦c≦N−1, where $\beta_{r} = \left\{ {\begin{matrix}{{1/\sqrt{2}},} & \; & {r = 0} \\{1,} & {{r = 1},2,\ldots\mspace{14mu},} & {N - 1}\end{matrix};} \right.$ and inserting a contiguous set of G zeros in thetime domain after the set of time domain samples and transmitting aresulting signal to produce a zero-padded sequence equal to {right arrowover (Y)}′(i)=C_(zp){right arrow over (Y)}(i) where C_(zp)=[I_(N)0_(N×G)]^(T) is an M× N zero-padding matrix, and I_(N) is an N×Nidentity matrix, and O_(N×G) is an N×G matrix containing all zeros. 9.The method of claim 8 further comprising: at a receiver, receiving theresulting signal over a channel, and during each OFDM symbol intervalprocessing the resulting signal received over the channel to recoverestimates of the data symbols.
 10. The method of claim 9 whereinprocessing comprises performing MMSE estimation.
 11. The method of claim8 wherein: the inverse discrete cosine transform is an N-point IDCT thatproduces a set of N time domain samples; inserting zeros comprisesinserting G zeros in the time domain after each set of N time domainsamples, where G is selected to substantially eliminate ISI betweenadjacent OFDM symbols after transmission over the channel.
 12. Themethod of claim 10 wherein the MMSE estimation is performed inaccordance with:{circumflex over (X)}(i)=DH ^(H)(HH ^(H)+σ² I _(M))⁻¹ {right arrow over(R)}(i)=D(H ^(H) H+σ ² I _(N))⁻¹ H ^(H) R(i)  (13) where I_(M) is an M×Midentity matrix, X(i) is the MMSE estimate, H=H₀C_(zp), where H₀ is achannel matrix, and C_(zp) is a zero padding matrix, D is a matrixrepresentation of the discrete cosine transform, M is a sequence lengthincluding zero padding, and N is a sequence length not including zeropadding, (*)^(H) is a conjugate transpose operation, and σ² is anadditive white noise Gaussian noise variance.
 13. The method of claim 9further comprising: performing one dimensional modulation anddemodulation.
 14. The system of claim 9 comprising: performing twodimensional modulation and demodulation.
 15. A receiver adapted to,during each OFDM symbol interval: receive a received version of atransmitted signal, the transmitted signal comprising a set of timedomain samples for each OFDM symbol interval in which data symbols aremapped to a set of N orthogonal sub-carriers of the form cos(2πnF_(Δ)t)where n=0, 1, . . . , N−1 and 0≦t<T, where N≧2, F_(Δ) is a frequencyspacing between subcarriers, T is a time duration of each OFDM symbolinterval, wherein the set of time domain symbols {right arrow over(Y)}(i) equal D^(T) X(i) where X(i) are the data symbols, and whereinD^(T) is the transpose of an N×N matrix with the element in the cth rowand the rth column$D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}}$for 0≦r≦N−1, and 0≦c≦N−1, where $\beta_{r} = \left\{ {\begin{matrix}{{1/\sqrt{2}},} & \; & {r = 0} \\{1,} & {{r = 1},2,\ldots\mspace{14mu},} & {N - 1}\end{matrix},} \right.$ and having a guard band of time with acontiguous set of G zeros inserted between consecutive symbol intervalsto produce a zero-padded sequence equal to {right arrow over(Y)}′=C_(zp){right arrow over (Y)}(i) where C_(zp)=[I_(N) 0_(N×G)]^(T)is an M× N zero-padding matrix, and I_(N) is an N×N identity matrix, andO_(N×G) is an N×G matrix containing all zeros, the guard band having atime duration selected to substantially eliminate ISI betweenconsecutive symbols after transmission over a channel; and during eachOFDM symbol interval, perform MMSE estimation to recover estimates ofthe data symbols.
 16. The receiver of claim 15 adapted to performdemodulation of a one dimensional modulation scheme.
 17. The receiver ofclaim 15 adapted to perform demodulation of a two dimensional modulationscheme.
 18. A receiver comprising: at least one antenna for receiving asignal, wherein the signal is a received version of a transmittedsignal, the transmitted signal comprising a set of time domain samplesfor each OFDM symbol interval in which data symbols are mapped to a setof N orthogonal sub-carriers of the form cos(2πnF_(Δ)t) where n=0, 1, .. . , N−1 and 0≦t<T, where N≧2, F_(Δ) is a frequency spacing betweensubcarriers, T is a time duration of each OFDM symbol interval, whereinthe set of time domain symbols {right arrow over (Y)}(i) equal D^(T)X(i) where X(i) are the data symbols, and wherein D^(T) is the transposeof an N×N matrix with the element in the cth row and the rth columndefined by$D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}}$for 0≦r≦N−1, and 0≦c≦N−1, where $\beta_{r} = \left\{ {\begin{matrix}{{1/\sqrt{2}},} & \; & {r = 0} \\{1,} & {{r = 1},2,\ldots\mspace{14mu},} & {N - 1}\end{matrix},} \right.$ and having a guard band of time with acontiguous set of G zeros inserted between consecutive symbol intervalsto produce a zero-padded sequence equal to {right arrow over(Y)}′(i)=C_(zp){right arrow over (Y)}(i) where C_(zp)=[I_(N)0_(N×G)]^(T) is an M× N zero-padding matrix, and I_(N) is an N×Nidentity matrix, and O_(N×G) is an N×G matrix containing all zeros, theguard band having a time duration selected to substantially eliminateISI between consecutive symbols after transmission over a channel; anestimator that during each OFDM symbol interval, recovers estimates ofthe data symbols from said signal by performing MMSE estimation inaccordance with: $\begin{matrix}\begin{matrix}{{\hat{X}(i)} = {{{DH}^{H}\left( {{HH}^{H} + {\sigma^{2}I_{M}}} \right)}^{- 1}{\overset{\rightharpoonup}{R}(i)}}} \\{= {{D\left( {{H^{H}H} + {\sigma^{2}I_{N}}} \right)}^{- 1}H^{H}{\overset{\rightharpoonup}{R}(i)}}}\end{matrix} & (13)\end{matrix}$ where I_(M) is an M×M identity matrix, {circumflex over(X)}(i) is the MMSE estimate, H=H₀C_(zp), where H₀ is a channel matrix,and C_(zp) is a zero padding matrix, D is a matrix representation of thediscrete cosine transform, M is a sequence length including zero paddingcomprising a contiguous set of zeros having a length selected tosubstantially eliminate intersymbol interference (ISI) betweenconsecutive OFDM symbols after transmission over a channel, and N is asequence length not including zero padding, (*)^(H) is a conjugatetranspose operation, and σ² is an additive white noise Gaussian noisevariance.
 19. A method comprising: receiving a signal over a channelwherein the signal is a received version of a transmitted signal, thetransmitted signal comprising a set of time domain samples for each OFDMsymbol interval in which data symbols are mapped to a set of Northogonal sub-carriers of the form cos(2πnF_(Δ)t) where n=0, 1, . . . ,N−1 and 0≦t<T, where N≧2, F_(Δ) is a frequency spacing betweensubcarriers, T is a time duration of each OFDM symbol interval, whereinthe set of time domain symbols {right arrow over (Y)}(i) equal D^(T)X(i) where X(i) are the data symbols, and wherein D^(T) is the transposeof an N×N matrix with the element in the cth row and the rth columndefined by$D_{r,c} = {\sqrt{\frac{2}{N}}\beta_{r}{\cos\left( \frac{\pi\;{r\left( {{2c} + 1} \right)}}{2N} \right)}}$for 0≦r≦N−1, and 0≦c≦N−1, where $\beta_{r} = \left\{ {\begin{matrix}{{1/\sqrt{2}},} & \; & {r = 0} \\{1,} & {{r = 1},2,\ldots\mspace{14mu},} & {N - 1}\end{matrix},} \right.$ and having a guard band of time with acontiguous set of G zeros inserted between consecutive symbol intervalsto produce a zero-padded sequence equal to {right arrow over(Y)}′(i)=C_(zp){right arrow over (Y)}(i) where C_(zp)=[I_(N)0_(N×G)]^(T) is an M× N zero-padding matrix, and I_(N) is an N×Nidentity matrix, and O_(N×G) is an N×G matrix containing all zeros, theguard band having a time duration selected to substantially eliminateISI between consecutive symbols after transmission over a channel;during an OFDM symbol interval processing the signal received over thechannel to recover estimates of data symbols; wherein processingcomprises performing MMSE estimation; wherein the MMSE estimation isperformed in accordance with: $\begin{matrix}\begin{matrix}{{\hat{X}(i)} = {{{DH}^{H}\left( {{HH}^{H} + {\sigma^{2}I_{M}}} \right)}^{- 1}{\overset{\rightharpoonup}{R}(i)}}} \\{= {{D\left( {{H^{H}H} + {\sigma^{2}I_{N}}} \right)}^{- 1}H^{H}{\overset{\rightharpoonup}{R}(i)}}}\end{matrix} & (13)\end{matrix}$ where I_(M) is an M×M identity matrix, {circumflex over(X)}(i) is the MMSE estimate, H=H₀C_(zp), where H₀ is a channel matrix,and C_(zp) is a zero padding matrix, D is a matrix representation of thediscrete cosine transform, M is a sequence length including zero paddingcomprising a contiguous set of zeros having a length selected tosubstantially eliminate intersymbol interference (ISI) betweenconsecutive OFDM symbols after transmission over a channel, and N is asequence length not including zero padding, (*)^(H) is a conjugatetranspose operation, and σ² is an additive white noise Gaussian noisevariance.